söndag 16 juni 2013

Essence of Dynamics 1

The dynamics of a physical system can typically be described as an initial value problem of finding a vector function U(t) depending on time t such that

dU/dt + A(U) = F for t > 0 with U(0) = G,

where, A(U) is a given vector function of U, F(t) is a given forcing and G is a given intial value at t = 0. In the basic case A(U) = A*U is linear with A = A(t) a matrix depending on time, which is also the linearized form of the system describing growth/decay of perturbations characterizing stable/unstable dynamics.

An essential aspect of the dynamics is the perturbation dynamics described by the linearized system which is determined by the eigenvalues of its linearization matrix A, assuming for simplicity that A is diagonalizable and independent of time:

Here Case 1. represents a dissipative system with exponential decay of perturbations in forward time making long time prediction possible, but backward time reconstruction difficult because of exponential growth of perturbations. This is the dynamics of a diffusion process, e.g. the spreading of a contaminant by diffusion or heat conduction.

Case 2. is the reverse with forward prediction difficult but backward reconstruction possible. This is the dynamics of a Big Bang explosion.

Case 3. represents turbulent flow with both exponential growth and decay giving rise to complex dynamics without explosion, with mean-value but not point value predictability in forward time. The picture above shows the turbulent flow around an airplane with mean-value quantities like drag and lift being predictable (in forward time). This case represents the basic unsolved problem of classical mechanics which is now being uncovered by computational methods including revelation of the secret of flight (hidden in the above picture).

Case 4 represents wave propagation with possibilities of both forward prediction and backward reconstruction, with the harmonic oscillator as basic case.

There is a further limit case with A non-diagonalizable with an incomplete set of eigenvectors for a multiple zero eigenvalue, with possibly algebraic growth of perturbations, a case arising in transition to turbulence in parallel flow.